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## Partially defined $\sigma$-derivations on semisimple Banach algebras

### Tom 190 / 2009

Studia Mathematica 190 (2009), 193-202 MSC: 46H40, 47B47, 46H15. DOI: 10.4064/sm190-2-7

#### Streszczenie

Let $A$ be a semisimple Banach algebra with a linear automorphism ${\sigma}$ and let $\delta\colon I\rightarrow A$ be a ${\sigma}$-derivation, where $I$ is an ideal of $A$. Then $\Phi(\delta)(I\cap{\sigma}(I) )=0$, where $\Phi(\delta)$ is the separating space of $\delta$. As a consequence, if $I$ is an essential ideal then the ${\sigma}$-derivation $\delta$ is closable. In a prime $C^*$-algebra, we show that every $\sigma$-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the ${\sigma}$-derivation expansion formula on zero products.

#### Autorzy

• Tsiu-Kwen LeeDepartment of Mathematics
National Taiwan University
Taipei 106, Taiwan
e-mail
• Cheng-Kai LiuDepartment of Mathematics
National Changhua University of Education
Changhua 500, Taiwan
e-mail

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